Embodiments of the present invention generally relate to systems and methods for determining a set of orthonormal polynomials from a complete set of non-orthogonal polynomials over a circular or noncircular pupil. More particularly, embodiments relates to systems, methods, and software for wavefront analysis in optical design and testing for wavefront representation.
An optical system for imaging or propagation of laser beams generally has a circular pupil. It is quite common to analyze the aberrations of such a system with Zernike circle polynomials, which are orthogonal across a unit circle. However, often the pupil is circular for on-axis point objects only. For off-axis point objects, the pupil can be elliptical or irregular because of vignetting. For large optical systems, the primary mirror may be segmented, where each segment is hexagonal. Similarly, high-power laser beams may have rectangular or square cross sections. The use of circle polynomials for such cases may not be appropriate. Thus, there is a need for polynomials that are orthogonal over noncircular pupils.
Since the circle polynomials form a complete set, in principle, the aberration function can be expressed in terms of them regardless of the shape of the pupil. However, such an expansion can take a large number of terms and the advantages of orthogonality may be lost. For example, the expansion coefficients may not be independent of each other and their values may change as the number of aberration terms changes. Moreover, the variance of the aberration may not be equal to the sum of the squares of the aberration coefficients. Moreover, their physical significance may not be of much use since they may not correspond to balanced aberrations.
In practice, the aberrations of a system can be determined either by tracing rays through it or by testing it interferometrically. Thus, the aberrations may be known only at a discrete set of points, and the coefficients thus obtained may be in error even for an axial point object, since the Zernike circle polynomials, which are orthogonal over the full circular region, may not be orthogonal over the discrete points of the aberration data set.
Therefore, in light of above, it would be desirable to provide improved methods, systems, and software for determining a set of orthonormal polynomials over noncircular pupils or the discrete data set and techniques for obtaining the orthonormal aberration coefficients. Embodiments of the present invention provide solutions for at least some of these needs.